Use Stoke's theorem to evaluate ∫∫ScurlF⋅dS?
Use Stoke's theorem to evaluate ∫∫ScurlF⋅dS where F(x,y,z)=−19yzi+19xzj+15(x^2+y^2)zk and S is the part of the paraboloid z=x^2+y^2 that lies inside the cylinder x^2+y^2=1, oriented upward.
Use Stoke's theorem to evaluate ∫∫ScurlF⋅dS where F(x,y,z)=−19yzi+19xzj+15(x^2+y^2)zk and S is the part of the paraboloid z=x^2+y^2 that lies inside the cylinder x^2+y^2=1, oriented upward.
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Verified answer
Since INT_S curlF.dS = INT_C F.dr and C closes the circle x^2+y^2= 1 OR closes the surface of the paraboloid , INT_S curl F dS over the paraboloid must be equal to INT curl F .k dA
Curl F ( k direction ) is ( 19z-(-19z)) = 38z at z= 1 curlF.k dA = 38 dA
INT_S curl F.dS ( parabolid) = INT_F.dr = INT_A curlF.kdA = 38A= 39*pi*1^2 = 39pi